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Persistence Curves: A canonical framework for summarizing persistence diagrams

arXiv 1904.07768 ↗First posted 16 Apr 2019cs.CG cs.CV math.ATcited by 33

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Abstract (arXiv)

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the textitPersistence Curves (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications---texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.

Key facts. Cited by 33 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Apr 2019.

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Citing papers per year: 2021: 4; 2022: 5; 2023: 6; 2024: 7; 2025: 5; 2026: 6.

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  1. 2022ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discoverycited by 13
  2. 2022Euler Characteristic Curves and Profiles: a stable shape invariant for big data problemscited by 8
  3. 2021A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysiscited by 5
  4. 2024EMP: Effective Multidimensional Persistence for Graph Representation Learningcited by 4
  5. 2021A computationally efficient framework for vector representation of persistence diagramscited by 3
  6. 2022Classification of Stochastic Processes with Topological Data Analysiscited by 3
  7. 2021A Topological Data Analysis Based Classifiercited by 2
  8. 2023Multiparameter Persistent Homology for Molecular Property Predictioncited by 2
  9. 2024Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistencecited by 2
  10. 2021Smart Vectorizations for Single and Multiparameter Persistencecited by 1
  11. 2023A fast topological approach for predicting anomalies in time-varying graphscited by 1
  12. 2023Vector Summaries of Persistence Diagrams for Permutation-based Hypothesis Testingcited by 1
21 more
  1. 2023A Topological Machine Learning Pipeline for Classificationcited by 1
  2. 2023SE(3)-Invariant Multiparameter Persistent Homology for Chiral-Sensitive Molecular Property Predictioncited by 1
  3. 2024Time topological analysis of EEG using signature theorycited by 1
  4. 2025Comparing the Effects of Persistence Barcodes Aggregation and Feature Concatenation on Medical Imagingcited by 1
  5. 2025CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizationscited by 1
  6. 2025Hierarchical Persistence Velocity for Network Anomaly Detection: Theory and Applications to Cryptocurrency Marketscited by 1
  7. 2026Topology-Aware Multiscale Mixture of Experts for Efficient Molecular Property Predictioncited by 1
  8. 2022Individual Topology Structure of Eye Movement Trajectoriescited by 0
  9. 2022Prediction of textrmCO2 Adsorption in Nano-Pores with Graph Neural Networkscited by 0
  10. 2023Topology-enhanced machine learning for speech signal processingcited by 0
  11. 2024A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagramscited by 0
  12. 2024A Vectorization Method Induced By Maximal Margin Classification For Persistent Diagramscited by 0
  13. 2024Sleep Staging from Airflow Signals Using Fourier Approximations of Persistence Curvescited by 0
  14. 2024Vectorization of Persistence Diagrams for Topological Data Analysis in R and Python Using TDAvec Packagecited by 0
  15. 2025Classification of Firn Data via Topological Featurescited by 0
  16. 2025A Novel Patch-Based TDA Approach for Computed Tomography Imagingcited by 0
  17. 2026G-LoG Bi-filtration for Medical Image Classificationcited by 0
  18. 2026The Topological Stability Index: A Variance-Based Measure for Persistence Barcodescited by 0
  19. 2026From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Featurescited by 0
  20. 2026Topological Data Analysis and Graph-Theoretic Approaches for Tennis Match Predictioncited by 0
  21. 2026Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagramscited by 0